Impossible Objects of Your Choice: Designing Any 3D Objects from a 2D Line Drawing

There is a class of line drawings, called impossible objects, that are perceived as 3D structures but are impossible to completely construct in 3D space. Sugihara [1] proposes a systematic method for creating a type of impossible objects. This method provides a way to judge the existence of possible three-dimensional (3D) coordinate for a line drawing, and a way to compute it, if possible. There are, however, some technical difficulties in using Sugihara's method. Firstly, Sugihara's method requires to introduce a large number of variables for use in a set of equations, which requires some intense labor for a designer of impossible objects when programming. Secondly, in theory, there are an infinite number of possible 3D coordinates for the same line drawing, but Sugihara's method can only determine one of them for a pre-defined set of parameters. In practice, a designer of impossible objects may also wish to arrange the 3D coordinates' undetermined degree of freedom for a given two-dimensional (2D) line drawing. Given these technical issues in Sugihara's method, we propose a new method to explore not just some but all 3D coordinates for a 2D line drawing that the designer can use at will. The proposed method requires both a minimal number of variables in its computation, resulting in it being computationally cheap, and less manual programming. Moreover, the proposed method provides a user interface that the designer can use to manually adjust the degree of freedom in the class of constructible impossible objects. This allows the designer to create impossible objects that reflect their tastes.